A cluster variety $U$ , as defined in [Reference Fock and GoncharovFG09], is a scheme constructed by gluing together a collection of algebraic tori $\text{Spec}\,\Bbbk [M]$ , called clusters, via certain birational automorphisms called mutations. A non-zero $f\in \unicode[STIX]{x1D6E4}(U,{\mathcal{O}}_{U})$ is called universally positive if its restriction to each cluster is a Laurent polynomial $\sum _{m\in M}a_{m}z^{m}$ with non-negative integer coefficients, and indecomposable or atomic if it cannot be written as a sum of two other universally positive functions.

Fock and Goncharov predicted [Reference Fock and GoncharovFG09, Conjecture 4.1] that the atomic functions on $U$ form an additive basis for $\unicode[STIX]{x1D6E4}(U,{\mathcal{O}}_{U})$ . However, [Reference Lee, Li and ZelevinskyLLZ14] showed this to be false by showing that the atomic functions are often linearly dependent, even in rank $2$ . Nevertheless, [Reference Gross, Hacking, Keel and KontsevichGHKK14] constructed a canonical topological basis of ‘theta functions’ $\{\unicode[STIX]{x1D717}_{m}\}_{m\in M}$ for a topological algebra $A$ that should be viewed as a formal version of (a base extension of) $\unicode[STIX]{x1D6E4}(U,{\mathcal{O}}_{U})$ ; e.g.  $A=\widehat{\text{up}(\overline{{\mathcal{A}}}_{\text{prin}}^{s})}\otimes _{\Bbbk [N_{s}^{+}]}\Bbbk [N]$ as in [Reference Gross, Hacking, Keel and KontsevichGHKK14, § 6] when $U$ is the ${\mathcal{A}}$ -variety, or $A=\widehat{B}$ as in [Reference MandelMan15, § 2.4]. These theta bases satisfy many of the properties predicted by [Reference Fock and GoncharovFG09] (e.g. universal positivity, being parametrized by $M$ ), and in many cases (when ‘the full Fock–Goncharov conjecture holds’), they extend to bases of $\unicode[STIX]{x1D6E4}(U,{\mathcal{O}}_{U})$ .

The construction of $\{\unicode[STIX]{x1D717}_{m}\}_{m\in M}$ (cf. [Reference Gross, Hacking, Keel and KontsevichGHKK14] for the details) involves a ‘scattering diagram’ $\mathfrak{D}$ in $M_{\mathbb{R}}:=M\otimes \mathbb{R}$ . This $\mathfrak{D}$ consists of a set of codimension $1$ ‘walls’ in $M_{\mathbb{R}}$ (with attached functions), the union of which forms the support of $\mathfrak{D}$ , denoted $\text{Supp}(\mathfrak{D})$ . Each $Q\in M_{\mathbb{R}}\setminus \text{Supp}(\mathfrak{D})$ determines an inclusion $\unicode[STIX]{x1D704}_{Q}$ of $A$ into a certain Laurent series ring $\widehat{R[M]}$ ; that is a localization of a completion of a base extension of $\Bbbk [M]$ , or, more precisely, $\widehat{R[M]}=\widehat{\Bbbk [\unicode[STIX]{x1D70E}]}\otimes _{\Bbbk [\unicode[STIX]{x1D70E}]}\Bbbk [M^{\prime }]$ , where $M^{\prime }$ is some lattice containing $M$ (e.g.  $M\oplus M^{\ast }$ for the cluster algebra with principal coefficients), $\unicode[STIX]{x1D70E}$ is some cone in $M^{\prime }$ , and $\widehat{\Bbbk [\unicode[STIX]{x1D70E}]}$ denotes the completion of $\Bbbk [\unicode[STIX]{x1D70E}]$ with respect to its maximal ideal. We say a non-zero $f\in A$ is universally positive with respect to the scattering atlas if for every $Q\in M_{\mathbb{R}}\setminus \text{Supp}(\mathfrak{D})$ , $\unicode[STIX]{x1D704}_{Q}(f)\in \widehat{R[M]}$ is a formal Laurent series with non-negative integer coefficients. Such an $f$ is called atomic with respect to the scattering atlas if it cannot be written as a sum of two other elements which are universally positive with respect to the scattering atlas.

To justify the terminology, recall that for $Q_{1},Q_{2}\in M_{\mathbb{R}}\setminus \text{Supp}(\mathfrak{D})$ , the $\unicode[STIX]{x1D704}_{Q_{i}}$ are related by a ‘path-ordered product’ (cf. [Reference Gross, Hacking, Keel and KontsevichGHKK14, § 1.1]). These generalizations of mutations are automorphisms of $\widehat{R[M]}$ that take a monomial $z^{m}$ to a formal positive rational function, by which we mean a quotient $f_{1}/f_{2}$ for two non-zero Laurent series $f_{1},f_{2}\in \widehat{R[M]}$ with non-negative integer coefficients. Hence, the charts $\operatorname{Spf}\widehat{R[M]}{\hookrightarrow}\operatorname{Spf}A$ induced by the $\unicode[STIX]{x1D704}_{Q}$ form a formal positive atlas on $\operatorname{Spf}A$ , i.e. a positive atlas in the sense of [Reference Fock and GoncharovFG09, Definition 1.1], except that the split algebraic tori there are replaced with the copies of the formal algebraic torus $\operatorname{Spf}\widehat{R[M]}$ , and the transition maps now preserve formal positive rational functions. This formal positive atlas is what we call the scattering atlas.

We note that when $U$ is a cluster ${\mathcal{A}}$ -variety, there is a subset of the chambers of $M_{\mathbb{R}}\setminus \text{Supp}(\mathfrak{D})$ , called the cluster complex, such that the corresponding charts of the scattering atlas are just the clusters (restricted to $\operatorname{Spf}A$ ). The scattering atlas may therefore be viewed as an enlargement of the positive atlas consisting of the clusters, which we call the cluster atlas. Note that the cluster atlas is the one considered by [Reference Fock and GoncharovFG09]. It is not clear in general whether the theta functions are indecomposable with respect to the cluster atlas, but [Reference Cheung, Gross, Muller, Musiker, Rupel, Stella and WilliamsCGM+17] has shown this for rank  $2$ .

The same argument proves that the quantum theta bases (as constructed in [Reference MandelMan15]) are exactly the atomic elements of the corresponding quantum algebras with respect to the scattering atlas, assuming universal positivity of these bases. This positivity fails in general but is expected for skew-symmetric cases, where positivity of the cluster variables in all clusters was recently proved in [Reference DavisonDav16].